Squad Selection ILP¶
Two-Level Formulation¶
The optimizer solves a single joint ILP for squad selection (Level 1) and lineup selection (Level 2). Binary variables \(s_i \in \{0,1\}\) indicate squad membership; \(x_i \in \{0,1\}\) indicate starting lineup membership.
Objective (risk-averse):
where \(\lambda \geq 0\) is the risk-aversion parameter and \(\rho_i\) is the risk measure (see Risk Penalties). Setting \(\lambda = 0\) gives the risk-neutral objective.
Level 1 — Squad constraints:
| Constraint | Expression |
|---|---|
| Squad size | \(\sum_i s_i = 15\) |
| Position quotas | \(s_{GK}=2\), \(s_{DEF}=5\), \(s_{MID}=5\), \(s_{FWD}=3\) |
| Budget | \(\sum_i c_i s_i \leq 100\) (applied to the full 15-player squad) |
| Team diversity | \(\sum_{i \in \text{team}_t} s_i \leq 3 \;\forall t\) |
Level 2 — Lineup constraints:
| Constraint | Expression |
|---|---|
| Lineup size | \(\sum_i x_i = 11\) |
| Subset of squad | \(x_i \leq s_i \;\forall i\) |
| GK | \(\sum_{i:p(i)=GK} x_i = 1\) |
| DEF | \(3 \leq \sum_{i:p(i)=DEF} x_i \leq 5\) |
| MID | \(2 \leq \sum_{i:p(i)=MID} x_i \leq 5\) |
| FWD | \(1 \leq \sum_{i:p(i)=FWD} x_i \leq 3\) |
Budget applies to the full squad
The £100m budget constraint covers all 15 players, not just the 11 starters. Formulations that apply the budget only to the lineup are incorrect and produce infeasible squads.
Risk Penalties¶
Both penalties keep the objective linear because \(x_i\) is binary and the risk coefficient \(\rho_i\) is computed by the inference pipeline before solving.
Mean-Variance Penalty¶
Penalizes all variance symmetrically. Tends to penalize star performers with high upside, so should be preferred only when downside risk estimates are unavailable.
Semi-Variance Penalty¶
Under a Gaussian predictive distribution, \(\rho_i^{\text{SV}} = \hat{\sigma}_i / \sqrt{2}\).
Penalizes only downside deviations, leaving upside variance intact. This is the
preferred risk measure and is returned directly by enriched_predict.
Backtest result: Semivar \(\lambda=0.5\) in 2024–25 reduces CV from 0.300 to 0.260 (−13%) at a cost of only −1.9% total points (1791 → 1757).
Downside Risk from TFT¶
When TFT quantile outputs are available, the spread \(\delta_i = q_{50,i} - q_{10,i}\) is used directly as \(\rho_i\). This is a distribution-free, data-driven risk measure.
Oracle Score¶
The oracle provides the performance ceiling for any forecasting system:
The same TwoLevelILPOptimizer is called with actual realized points as
inputs. This is not the sum of the top-11 scorers — budget, position
quotas, and team caps are all enforced.
Oracle averages 138.2 pts/GW (2023–24) and 132.4 pts/GW (2024–25). The best strategy (ILP + Enriched) achieves ~42% of oracle.
Double Gameweek Handling in the ILP¶
aggregate_dgw_timeseries (called automatically in the data layer) ensures
inference always outputs per-fixture expected points. Before the optimizer
call, scale_predictions_for_dgw applies the multiplier:
| Quantity | Scaling | Rationale |
|---|---|---|
| \(\hat{\mu}_i\) | \(\times n_i\) | EP adds linearly under match independence |
| \(\hat{\sigma}^2_i\) | \(\times n_i\) | Variance adds under independence |
| \(\hat{\sigma}^-_i\) | \(\times \sqrt{n_i}\) | Semi-deviation scales as \(\sqrt{n}\) |
| BGW player (\(n_i=0\)) | \(\hat{\mu}_i = 0\) | No fixture; naturally excluded by ILP |
DGW players (\(n_i=2\)) receive doubled expected points as their objective coefficient and become the most attractive picks — which is correct.
See Double Gameweeks for the full design.
Lagrangian Relaxation¶
The budget constraint is dualized with multiplier \(\mu \geq 0\):
The dual \(D(\mu) = \min_{\mu \geq 0} L(\mu)\) is minimized via subgradient ascent. At each iteration, the unconstrained inner problem decomposes into independent per-player problems solved greedily. Subgradient: \(g_k = B - \sum_i c_i s^*_i(\mu_k)\).
Backtest result: Lagrangian relaxation recovers 97.7% (2024–25) and 98.1% (2023–24) of full ILP performance, converging in ~50 iterations.
Valid Formations¶
The lineup constraints admit 8 valid formations: 3-4-3, 3-5-2, 4-3-3, 4-4-2, 4-5-1, 5-2-3, 5-3-2, 5-4-1. Formation is determined automatically by the optimizer — no manual selection needed.
Usage¶
from fplx.selection.optimizer import TwoLevelILPOptimizer
optimizer = TwoLevelILPOptimizer(budget=100.0, risk_aversion=0.5)
squad = optimizer.optimize(
players=players,
expected_points=ep,
expected_variance=var, # for mean-variance penalty
downside_risk=downside_risks, # overrides variance if provided
)
# squad.squad_players -> 15 Player objects
# squad.lineup.players -> 11 starters
# squad.lineup.captain -> highest-EP player